Problem: Find the point on the line
\[y = \frac{x + 5}{2}\]that is closest to the point $(6,1).$
Explanation: Note that $(1,3)$ and $(3,4)$ are two points on the line, so the line has a direction vector of
\[\begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}.\][asy]
unitsize(0.5 cm);

pair A, B, C, D, V, P;

A = (-8, (-8 + 5)/2);
B = (5, (5 + 5)/2);
C = (1,3);
D = (3,4);
V = (6,1);
P = (V + reflect(A,B)*(V))/2;

draw((-8,0)--(8,0));
draw((0,-4)--(0,5));
draw(A--B,red);
draw(V--P,dashed);
draw(C--V,Arrow(6));
draw(C--D,Arrow(6));

dot("$(6,1)$", V, E);
dot("$(1,3)$", C, NW);
dot("$(3,4)$", D, NW);
[/asy]

The vector going from $(1,3)$ to $(6,1)$ is $\begin{pmatrix} 6 \\ 1 \end{pmatrix} - \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}.$  Projecting this vector onto the direction vector, we get
\[\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \frac{\begin{pmatrix} 5 \\ -2 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix}}{\left\| \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\|^2} \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \frac{8}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} \frac{16}{5} \\ \frac{8}{5} \end{pmatrix}.\][asy]
usepackage("amsmath");

unitsize(0.5 cm);

pair A, B, C, D, V, P;

A = (-8, (-8 + 5)/2);
B = (5, (5 + 5)/2);
C = (1,3);
D = (3,4);
V = (6,1);
P = (V + reflect(A,B)*(V))/2;

draw((-8,0)--(8,0));
draw((0,-4)--(0,5));
draw(A--B,red);
draw(V--P,dashed);
draw(C--V,Arrow(6));
draw(C--P,Arrow(6));

label("$\begin{pmatrix} \frac{16}{5} \\ \frac{8}{5} \end{pmatrix}$", P, NW);

dot("$(6,1)$", V, E);
dot("$(1,3)$", C, NW);
[/asy]

Then
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + \begin{pmatrix} \frac{16}{5} \\ \frac{8}{5} \end{pmatrix} = \begin{pmatrix} \frac{21}{5} \\ \frac{23}{5} \end{pmatrix},\]so the point on the line closest to $(6,1)$ is $\boxed{\left( \frac{21}{5}, \frac{23}{5} \right)}.$